Measure-theoretic aspects of Convex bodies

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。拟议研究的对象是高维凸体或更一般的对数凹概率测度的质量分布的研究。最近，概率论的几个经典结果已被扩展到更广泛的设置这些措施。 这些新的结果似乎是遥不可及的经典概率推理的基础上的独立性，取代了几何概念的凸性。PI打算进一步研究高维测度的几何参数，特别是 广义质心机构与这些措施。有了这种方法和应用技术从局部理论的Banach空间，经典凸性，概率和信息论，PI希望攻击各种开放的问题有关的日志凹措施，如小球概率估计和超平面猜想，以及各种programmures在理论的凸体，如规律性的熵数和最佳界的最小平均宽度。有一个普遍的原则，在这个建议的研究基础：高维系统的趋势聚集在典型的形式。这是一个中心的事实，影响研究的复杂系统出现在概率，组合，统计物理和复杂性。高维凸体作为概率空间时的意外规则性，有望为我们理解这一一般原理增加一个新的组成部分。 随机多面体，随机矩阵和随机算法（分别来自组合学，数学物理学和信息学）的应用已经被发现，这表明未来应该有更多的应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The object of the proposed research is the study of the distribution of mass of high dimensional convex bodies or more general of log-concave probability measures. Recently, several classical results from probability theory have been extended to the broader setting of these measures. These new results appear to be out of reach of the classical probabilistic reasoning based on independence which is replaced by the geometric notion of convexity. The PI intends to further investigate the geometric parameters of high-dimensional measures and in particular the geometry of generalized centroid bodies associated to these measures. With this approach and applying techniques from local theory of Banach spaces, classical convexity, probability and information theory, the PI wishes to attack various open problems related to log-concave measures such as small ball probability estimates and the hyperplane conjecture, as well as, various conjectures in the theory of convex bodies such as the regularity of the entropy numbers and the optimal bound of the minimal mean width. There is a general principle that underlies the research in this proposal: the tendency of high dimensional systems to congregate around typical forms. This is a central fact that influences the study of complex systems which appear in probability, combinatorics, statistical physics and complexity. The unexpected regularity of high dimensional convex bodies when viewed as probability spaces is expected to add a new component to our understanding of this general principle. Applications to random polytopes, random matrices, and random algorithms (topics from combinatorics, mathematical physics and informatics respectively) have already been discovered, indicating thatmore should be expected in the future.